Given the rectangle ABCD has a total area of 72. E is in the midpoint of BC and F is the midpoint of DC. What is the area of the inscribed triangle AEF?
To find the area of the inscribed triangle AEF, we need to determine the lengths of the sides AE and AF first.
Let's analyze the rectangle ABCD. Since E is the midpoint of BC and F is the midpoint of DC, we can deduce that EF is a diagonal of the rectangle. The diagonal EF divides the rectangle into two congruent right triangles AEF and CEF.
Now let's calculate the lengths AE and AF. We know that E is the midpoint of BC, which means AE is half the length of BC. Similarly, F being the midpoint of DC implies AF is half the length of DC.
Since the rectangle has an area of 72, we can find the lengths of BC and DC by factoring the area. Let's find the factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.
From the factors, we observe that the pair 8 and 9 fits our requirements since 8 * 9 = 72. Therefore, the lengths of BC and DC are 9 and 8 respectively.
Now, we can calculate the lengths of AE and AF:
- AE = BC / 2 = 9 / 2 = 4.5
- AF = DC / 2 = 8 / 2 = 4
Finally, to find the area of triangle AEF, we can use the formula for the area of a triangle which is A = (base * height) / 2.
In our case, the base is 4.5 and the height is 4 since AE and AF are the base and height respectively. Plugging these values into the formula, we get:
Area of AEF = (4.5 * 4) / 2 = 18 / 2 = 9
Therefore, the area of the inscribed triangle AEF is 9 square units.
To find the area of the inscribed triangle AEF, we need to find the lengths of its sides first.
Since E is the midpoint of BC, we can say that AE is equal to EC. Similarly, as F is the midpoint of DC, AF is equal to FD.
Let's denote the length of AE and EC by x, and the length of AF and FD by y.
Since ABCD is a rectangle, the length of AB is equal to the length of CD, and the length of AD is equal to the length of BC.
Given that the total area of the rectangle is 72, we can set up the equation:
AB * AD = 72
Since AB is equal to CD and AD is equal to BC, we can rewrite the equation as:
CD * BC = 72
Since E is the midpoint of BC, BC is equal to 2x. Similarly, FD = AF = y.
Substituting these values into the equation, we get:
CD * 2x = 72
Simplifying the equation:
2 * CD * x = 72
CD * x = 36
Since CD is equal to AB, we have:
AB * x = 36
We know that AB is the length of the base of the triangle AEF, and x is the height of the triangle.
Therefore, the area of the triangle AEF is:
Area = (1/2) * AB * x
Substituting the value of AB * x from the equation above, we get:
Area = (1/2) * 36
Area = 18
So, the area of the inscribed triangle AEF is 18.
If BC=2y and CD=2x, note that triangles
ADF, ECF, and ABE are all right triangles with one leg half of a side of the rectangle. We know that ABCD has area (2x)(2y) = 72, so xy=18.
ADF has area (2y)(x)/2 = xy = 18
ECF has area (y)(x)/2 = xy/2 = 9
ABE has area (y)(2x)/2 = xy = 18
So, AEF has area 72-18-18-9 = 27